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I have n sectors, enumerated 0 to n-1 counterclockwise. The boundaries between these sectors are infinite branches (n of them).

These branches meet at certain points (junctions). Each junction is adjacent to a subset of the sectors (at least 3 of them).

By specifying what sectors my junctions are adjacent to, I can completely recover the tree. This seems like something known, but I would like a reference to it.

The number of trees with n branches is given by http://www.research.att.com/~njas/sequences/A001003 and this is quite easy to prove.

Furthermore, if I order the sectors in the description of the junctions, I can make this representation unique.

Example: (0,1,2,3,4,5) represents the tree with only one vertex, and 6 branches connected to this junction.

flag	asked Feb 17 2010 at 0:04 Per Alexandersson 1,846●6●12

Perhaps it's just me, but I'm completely unable to parse this. Could you perhaps explain what you mean by sectors, infinite branches, junctions and what is the tree? A picture, perhaps? – Alon Amit Feb 17 2010 at 1:11

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This is almost, but not quite, what a Prufer code does: en.wikipedia.org/wiki/Pr%C3%BCfer_sequence – Qiaochu Yuan Feb 17 2010 at 3:20

I cannot post more hyperlinks since i am a new user, Here are 4 examples of such trees (ignore vertices with degree 2 or less) www2.math.su.se/~per/bloggimages/… www2.math.su.se/~per/bloggimages/… – Per Alexandersson Feb 17 2010 at 16:51

Per Alexandersson · Accepted Answer · 2010-02-17 17:40:01Z UTC

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I realized that taking the dual of my trees, I always get an n-gon, where some chords, the faces in the dual are my junctions. The bijection is now trivial.

answered Feb 17 2010 at 17:40

Per Alexandersson
1,846●6●12

Describe a tree by junctions

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